$ W^{2, p} $-regularity for asymptotically regular fully nonlinear elliptic and parabolic equations with oblique boundary values

Junjie Zhang; Shenzhou Zheng; Chunyan Zuo

doi:10.3934/dcdss.2021080

Article Contents

2021, Volume 14, Issue 9: 3305-3318. Doi: 10.3934/dcdss.2021080

This issue Previous Article Existence and multiplicity of positive solutions for a class of quasilinear Schrödinger equations in $ \mathbb R^N $$ ^\diamondsuit $ Next Article Non-autonomous weakly damped plate model on time-dependent domains

$ W^{2, p} $-regularity for asymptotically regular fully nonlinear elliptic and parabolic equations with oblique boundary values

1.
School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, Hebei 050016, China
2.
Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China
3.
School of Sciences, Hebei University of Science and Technology, Shijiazhuang, Hebei 050016, China

^* Corresponding author: Shenzhou Zheng
^* Corresponding author: Shenzhou Zheng

Received: February 29, 2020

Revised: December 31, 2020

Early access: June 2021

Published: September 2021

The first author is supported by National Natural Science Foundation of China (No. 12001160), Natural Science Foundation of Hebei Province (No. A2019205218), Science Foundation of Hebei Normal University (No. L2019B02). The second author is supported by National Natural Science Foundation of China (No. 12071021)

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Abstract

We prove a global $ W^{2, p} $-estimate for the viscosity solution to fully nonlinear elliptic equations $ F(x, u, Du, D^{2}u) = f(x) $ with oblique boundary condition in a bounded $ C^{2, \alpha} $-domain for every $ \alpha\in (0, 1) $. Here, the nonlinearities $ F $ is assumed to be asymptotically $ \delta $-regular to an operator $ G $ that is $ (\delta, R) $-vanishing with respect to $ x $. We employ the approach of constructing a regular problem by an appropriate transformation. With a similar argument, we also obtain a global $ W^{2, p} $-estimate for the viscosity solution to fully nonlinear parabolic equations $ F(x, t, u, Du, D^{2}u)-u_{t} = f(x, t) $ with oblique boundary condition in a bounded $ C^{3} $-domain.

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Mathematics Subject Classification: Primary: 35B65, 35J25, 35K20.

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